Deviations from Linearity

Recall that p, the parameter that characterizes a particular reaction in a Hammett analysis, is the slope obtained from the plot of x/ h vs. a. The plot must give a straight line to get a p value. What does it mean if a Hammett plot is not linear  

Using a Hammett Plot to Explore the Behavior of a Catalytic Antibody 

Recently, a new way to develop catalysts has emerged known as the study of catalytic antibodies. Herein, transition state analogs for various reactions are created, and antibodies are isolated that bind these analogs. As we will see in Chapter 9, the binding of the transition state of a reaction is the major factor by which catalysis is achieved with enzymes. Thus, the antibodies created to bind a transition state analog are often found to catalyze the reaction involving the transition state the analog was patterned after. 

Closed circles are the data for the antibody-catalyzed pathway. The diamonds are the data for the reaction in solution. 

Wentworth, P., Jr., Datta, A., Smith, S., Marshall, A., Partridge, L. J and Blackburn, G. M. Antibody Catalysis of Aryl Carbamate Ester Hydrolysis A Highly Disfavored Chemical Process. /. Am. Chem. Soc., 119,2315-2316(1997). 

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Frequently, however, the DR plot deviates from linearity, and in a number of ways. Sometimes the plot is convex to the log (p°/p) axis, as in Fig. 4.19(a), and sometimes concave, as in Fig. 4.19(b). The question then arises as to whether one should extropolate from the low-pressure. 

According to Beer s law, a calibration curve of absorbance versus the concentration of analyte in a series of standard solutions should be a straight line with an intercept of 0 and a slope of ab or eb. In many cases, however, calibration curves are found to be nonlinear (Figure 10.22). Deviations from linearity are divided into three categories fundamental, chemical, and instrumental. 

The particle sizes of fillers are usually collected and ordered to yield size distributions which are frequendy plotted as cumulative weight percent finer than vs diameter, often given as esd, on a log probabiUty graph. In this manner, most unmodified fillers yield a straight-line relationship or log normal distribution. Inspection of the data presented in this manner can yield valuable information about the filler. The coarseness of a filler is often quantified as the esd at the 99.9% finer-than value. Deviations from linearity at the high and low ends of the plot suggest that either fractionation has occurred to remove coarse or fine particles or the data are suspect in these ranges. 

Because of the geometric limitations of the indenting ball the relationship between indentation area and computed hardness number deviates from linearity when the recovered indentation diameter of a 10-mm ball is less than 2.5 mm or greater than 6.0 mm. 

For many modeling purposes, Nhas been assumed to be 1 (42), resulting in a simplified equation, S = C, where is the linear distribution coefficient. This assumption usually works for hydrophobic polycycHc aromatic compounds sorbed on sediments, if the equdibrium solution concentration is <10 M (43). For many pesticides, the error introduced by the assumption of linearity depends on the deviation from linearity. 

For intermediate drift rates (4 < BN < 8), when chain conformations are already distorted, deviates from linear behavior and goes through a maximum at some critical value Bf. of the field, confirming earlier findings by Pandey et al. [103,104]. This critical bias B at which the velocity starts to decrease depends rather weakly on the density Cobs, turns out to be reciprocal to chain length A, implying that only when the total force, /c = B,N 9, acting upon the whole driven molecule, exceeds a certain threshold, which does not depend on the size of the macromolecule, the chains start to get stuck in the medium. 

Figure 2-11, Semilogarithmic plot of the hydrolysis of p-nitrophenyl glutarate in the presenee of p-methoxycinnamate ion 25.0°C, initial pH 7.53, reaction followed at 400 nm. The plot deviates from linearity after the first half-life.

If the kinetics of the reaction disobey the Michaelis-Menten equation, the violation is revealed by a departure from linearity in these straight-line graphs. We shall see in the next chapter that such deviations from linearity are characteristic of the kinetics of regulatory enzymes known as allosteric enzymes. Such regulatory enzymes are very important in the overall control of metabolic pathways. 

Equation (4) indicates that the slope n of the log D versus the pH plot corresponds to the number of protons released upon extraction. If the logarithm of the ratio between Fe content in aqueous phase and organic phase is plotted as a function of pH, a linear relation was obtained between pH 3.5-5.4, which deviated from linearity at lower pH values (2.2). The fact that the slope of the curves were very close to unity indicates that only one proton has separated from the ligand (Eq. [5]). 

The amount of fretting damage increases in an approximately linear manner with these variables, once the Initial stages of fretting are completed. A number of deviations from linearity have been reported, especially with respect to load, where it is often found that there is a tendency for the relationship to become parabolic in form. The superposition of a normal vibratory component of load can cause a very considerable increase in the wear rate ... 

The importance of the method in corrosion testing and research has stimulated other work, and since Stern s papers appeared there have been a number of publications many of which question the validity of the concept of linear polarisation. The derivation of linearity polarisation is based on an approximation involving the difference of two exponential terms, and a number of papers have appeared that have attempted to define the range of validity of polarisation resistance measurements. Barnartt" derived an analytical expression for the deviations from linearity and concluded that it varied widely between different systems. Leroy", using mathematical and graphical methods, concluded that linearity was sufficient for the technique to be valid in many practical corrosion systems. Most authors emphasise the importance of making polarisation resistance measurements at both positive and negative overpotentials. 

FIGURE 6.15 Example of application of method of Lew and Angus [10]. (a) Dose-response data, (b) Clark plot according to Equation 6.27 shown, (c) Data refit to power departure version of Equation 6.27 to detect slopes different from unity (Equation 6.28). (d) Data refit to quadratic departure version of Equation 6.27 to detect deviation from linearity (Equation 6.29). 

In loading experiments the separation of the matrix from the filler is one of the reasons responsible for the deviation of the stretching diagram filler concentration if adhesion is poor, it remains invariant up to a certain concentration and then drops very suddenly. 

Even under random coil conditions at rapid cooling and instant measuring, no linear temperature function can be observed. The deviation from linearity is the higher, the more the temperature decreases and the chain length increases. 

Fig. 19. Example of deviation from linearity between burn-out heat flux and inlet subcooling [from Lee and Obertelli (L6)]. Based on tests with water in a uniformly heated tube, L = 34 in., d = 0.426 in., P = 1000 psia.

The magnitude of t0 can be measured from the intercept of a f(a)—time plot. The existence of the induction period can introduce uncertainty into a reduced time analysis if the temperature coefficient of t0 differs from that later applicable, and it is necessary to plot (t — t0)/(tb — t0) against a where tb is the time at which the selected common value of a is attained. The occurrence of a slow initial process can be reflected in deviations from linearity in the f(a) time plot, though in favourable systems the contribution may be subtracted before analysis [40]. 

An increase in co from 400 to 1600rpm tints results in a twofold increase of the signal. A deviation from linearity of a plot of z) vs. col/1 suggests some kinetic limitations. In addition, at veiy low rotation speeds (0-100 rpm), a slight upward bend is observed due to contribution by natural convection. The voltammetric wave has a sigmoidal shape for reversible systems it is identical to that common in DC polarography (described in Section 3-2), and independent of to. 

In the radiolysis of diphenyl sulfone, the yield of S02 shows marked deviation from linearity as a function of the dose. For higher doses it appears that S02 yield is 2-3 times less than the biphenyl yield of 0.05. The authors suggested that not all of the S02 was detected and that a significant amount was trapped in the irradiated crystals. After... 

Parsons-Zobel plots show a remarkable curvature at cel < 0.05 M KF (Table 16). At c < 0.01 M, a more remarkable deviation from linearity has... 

Fedorenko S. G., Burshtein A. I. Deviations from linear Stern-Volmer law in hopping quenching theory, J. Chem. Phys. 97, 8223-32 (1992). 

The parameter aH is positive for electropositive (electron donor) adsorbates and negative for electronegative (electron acceptor) adsorbates. Even when deviations from linearity exist, the main feature of Eq. (2.23) remain valid and form the basis for understanding the main kinetic features of classical and electrochemical promotion ... 

The temperature-factor parameter B and the scale factor k were determined by a least-squares procedure/ with observational equations set up in logarithmic form and with weights obtained from those in equation (9) by multiplying by (G (obs.))2. Since a semi-logarithmic plot of G2 (obs.)/Gf (calc.) against B showed a pronounced deviation from linearity for the last five lines, these lines were omitted from the subsequent treatments. They were much broader than the others, and apparently their intensities were underestimated. The temperature-factor parameter B was found by this treatment to have the value 1-47 A2. 

The most remarkable structural features of the planar cyclic mixed trisethyne-monobutadiyne pentaacetylenes 181 and 62 are their drastically bowed diyne moieties [18]. The acetylenic carbon atoms deviate from linearity by an average of 11.7° in 181 and even 13.4° in 62. The internal C-C-C angle is smaller for 181 (103.8°) than for 62 (109.2°). The disparity of the C-C bond lengths in the spirocyclopropane moieties of 62 is analogous to those observed in the expanded [n]rotanes 165-168. 

The resistance of an organism to a sterilizing agent can be described by means of the D-value. For heat and radiation treatments, respectively, this is defined as the time taken at a fixed temperature or the radiation dose required to achieve a 90% reduction in viable cells (i.e. a 1 log cycle reduction in survivors Fig. 20.2k). The calculation of the D-value assumes a linear type A survivor curve (Fig. 20.1), and must be corrected to allow for any deviation from linearity with type B or C curves. Some typical D-values for resistant bacterial spores are given in Table 23.2 (Chapter 23). 

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